Let \(f\colon G\rightarrow H\) , be a continuous homomorphism into a topological group \(H\) .
By
[1]}, any non-trivial normal subgroup is either co-compact or central and finite.
If the kernel of \(f\)  is co-compact then the image \(f(G)\)  is compact and hence closed.
Thus we may assume that \(\ker (f)\le G\)  is a finite central subgroup. Let \(\hat{f}\colon G/\ker (f) \rightarrow H\)  be the injective map induced by \(f\) .
By Proposition REF  the group \(G/\ker (f)\)  is minimal hence \(f(G)=\hat{f}( G/\ker (f) )\)  is closed in \(H\) .
